Wagner mercenaries have fought alongside regular Russian troops in Ukraine, including in some of the most intense battles.
The Kremlin has dismissed the allegations of war crimes and arrest warrant by the ICC, saying The Court warrant holds no water over Russia.
Ms Zakharova accused authorities at the United Nations of preventing an inspection by the International Atomic Energy Agency (IAEA).
Russia has reiterated its warning of a possible failure of an agreement on ending a blockade of grain exports.
U.S. citizens whose visas for Russia are expiring have been asked to leave the country by June 15.